Whoa. That's some big plans. Where exactly do you live? That one bit of information will help us know why you are planning such an elaborate process.

I live in the Deep South. Here, we put our tulip bulbs into the fridge (away from ripening fruit: ethylene gas, you know. And yes, my experience says it definitely can have an effect.) for eight to ten weeks, then plant them outside in the ground, or in pots, either way. They come up in the spring and bloom and that's that.

Further north (U.S.A), they simply plant them in the fall, where they will get plenty of winter chill time before they come up and bloom in the spring. No soaking. No cutting.

Again. We really can't advise you until we know where you live.

I live in Lithuania (Europe), it's zone 5, I think.
Tulips are planted here into the ground at the end of September and they bloom at about the end of April.

"Tulip forcing with the 5*C method.

After keeping dry bulbs in temperatures of 5*C for 9-14 weeks, after the bloom completely forms (G stage), planted, the roots and the shoots grow simultaneously. Bulbs forced by this method are planted into greenhouse beds, and grown warmly right away. The plants bloom after about 6-10 weeks.
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To force with this method, only 4-5 cm bulbs are chosen, and they're planted 2 cm deep (counting from the top)."
(this is from the book)
I think I stated incorrectly, that I was going to keep dry bulbs for 8 weeks in the fridge, soak, plant into containers, and put back into the fridge for 8 more weeks, did I? And I shouldn't keep their "noses" (I mean bulb tips) out of the soil?

I found this page http://www.flowerbulbs.cornell.edu/forcing/pot_tulips.htm
It's research on varieties of tulips that are best for pots.
But now I'm getting more confused as I learn more...
First of, I'm now thinking about only storing dry bulbs in the fridge, not potted bulbs, BUT this website says that for most varieties 16 weeks is the optimal time, but the book says that the dry bulbs have to be stored for 9-14 weeks in 5*C. I'm assuming in that website they don't mean dry bulbs, they mean planted bulbs.
So this makes me still want to try to cool planted bulbs... But I don't understand if 16 weeks after they root, or 16 weeks with the rooting?

Can you buy 'prepared' bulbs in Lithuania? These have had all the chilling done for you by the bulb company and are ready to plant in the containers. You then put the containers in a cool frost free place until the shoots show and then bring them indoors.

Check Kevin Lee Jacob's blog called A garden for The House.
Look on the sidebar to 2009 and scan down his pictures, and you'll see he has a spare refrigerator in his house that he has potted up his bulbs and keeps them there. While you are there, browse over his entire website. It's full of very valuable information regarding flowers, cooking, deocorating, etc. From some postings, he writes for several gardening magazines. He knows his stuff!